In this paper ,we discuss the derivations of the symplectic ternaryalgebras,the correspondence between the trace form of a Lie triplesystem and that of a symplectic ternary algebra ,and the wedderburnprincipal theorem .

Using the trace form, we prove the existence of the complete set of primitive central idempotents of semi-simple algebra A on the field K of characteristic zero and exprese A as a direct sum of a finite number of irreducible submodules, further obtain a series of essential results of ordinary representation of finite groups. This method is shorter and clearer than the conventional methods ([1] or[2]).

Let L is a finite dimensional simple Z graded Lie superalgebra. In this paper, we obtain the necessary conditions that L possesses a nondegenerate trace form. Using this result, we obtain that the Killing forms of Lie superalgebras X(n) and X(m,n,t) of Cartan type are degenerate.

Specifically, we present a characterization of the Kuz'min radical in terms of a trace form associated with some representation ρ, which is analogous to the characterization which we have in the case of Lie algebras.

We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its r-linear trace form, $$ (a_1, \ldots, a_r) \mapsto tr(a_1, \ldots a_r), $$ for any r ≥ 3.

In this article, we show that the second trace form of a central simple algebra A of even degree over a field of characteristic two is non-degenerate and we compute its classical invariants.

If K has characteristic not two, it is shown in [U] that ?2,A does not give much more information than the usual trace form.

Let ?A and ?2,A be respectively the trace form and the second trace form of A.

Some fundamental theoretical problems in 4--dimension descriptive geometry, such as the projection law of a trace--form plane, the characteristic about projection position of geometrical elements etc, have not been perfected yet and should be discussed further. In this paper, after analyzing those characteristics mentioned above, 14 propositions about the representations of general positions of trace--form planes and semiparallel--semi perpendicular planes are presented. Based on these...

Some fundamental theoretical problems in 4--dimension descriptive geometry, such as the projection law of a trace--form plane, the characteristic about projection position of geometrical elements etc, have not been perfected yet and should be discussed further. In this paper, after analyzing those characteristics mentioned above, 14 propositions about the representations of general positions of trace--form planes and semiparallel--semi perpendicular planes are presented. Based on these results, a new method for the classification of various positions between planes and for the study of projection characteristic is given.

Using the trace form, we prove the existence of the complete set of primitive central idempotents of semi-simple algebra A on the field K of characteristic zero and exprese A as a direct sum of a finite number of irreducible submodules, further obtain a series of essential results of ordinary representation of finite groups. This method is shorter and clearer than the conventional methods ([1] or[2]).